The solidification of a dilute bismuth-tin alloy under Bridgman crystal growth conditions is investigated in support of NASA's MEPHISTO space shuttle flight experiment. Computations are performed in two-dimensions with a uniform grid. The simulation includes the species-concentration, temperature and flow fields, as well as conduction in the ampoule. Fully transient simulations have been performed; no simplifying steady state approximations are used. Results are obtained under microgravity conditions for pure bismuth, and Bismuth-0.1 at.% Sn and Bi-1.0 at.% Sn alloys. The concentration dependence of the melting temperature is neglected; the solid/liquid interface temperature is assumed to be the melting temperature of pure bismuth for all cases studied. For the Bi-1.0 at.% Sn case the results indicate that a secondary convective cell, driven by solutal gradients, forms near the interface. The magnitude of the velocities in this cell increases with time; this causes increasing solute segregation at the liquid/solid interface.